Change in types of neuronal excitability via bifurcation control

This paper proposes an approach to changing the types of neuronal excitability via bifurcation control. A washout filter-aided dynamic feedback controller is introduced to bifurcation dynamics of a two-dimensional Hindmarsh-Rose type model neuron, which shows a saddle-node on invariant circle (SNIC) bifurcation from quiescence to periodic spiking and then exhibits type-I excitability. At first, a Hopf bifurcation is created at a desired parameter value before the SNIC bifurcation occurs, and then the criticality of the created Hopf bifurcation is regulated by choosing appropriate values of the controller parameters. In this manner, the model neuron starts to show type-II excitability. Therefore the type of neuronal excitability is transformed from type-I excitability to type-II excitability for the model neuron via the washout filter-aided dynamic feedback controller. In such a controller, the linear control gain is determined by the two basic critical conditions for the Hopf bifurcation, i.e., the eigenvalue assignment and the transversality condition. We apply the center manifold and normal form theory to deduce a closed-form analytic expression for the bifurcation stability coefficient, which is a function with respect to the nonlinear control gain. A suitable nonlinear control gain is chosen to make the bifurcation stability coefficient negative, and thus the criticality of the created Hopf bifurcation can be changed from subcritical to supercritical. In addition, the amplitude of the corresponding periodic solution can be also regulated by the nonlinear control gain.